We extend Geisser and Hesselholt's result on "bi-relative Ktheory" from discrete rings to connective ring spectra. That is, if A is a homotopy cartesian n-cube of ring spectra (satisfying connectivity hypotheses), then the (n + 1)-cube induced by the cyclotomic traceis homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace K(A) → T C(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision.The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and -more relevantly for our current application -the T-Tate spectrum of topological Hochschild homology, where T is the circle group.

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